Theorem imaeq2 5005

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4941	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4895	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4676	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4676	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1364   ran crn 4664   |` cres 4665   "cima 4666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  imaeq2i  5007  imaeq2d  5009  fimadmfo  5489  ssimaex  5622  ssimaexg  5623  isoselem  5867  f1opw2  6129  fopwdom  6897  ssenen  6912  fiintim  6992  fidcenumlemrk  7020  fidcenumlemr  7021  sbthlem2  7024  isbth  7033  ennnfonelemp1  12623  ennnfonelemnn0  12639  ctinfomlemom  12644  ctinfom  12645  tgcn  14444  iscnp4  14454  cnpnei  14455  cnima  14456  cnconst2  14469  cnrest2  14472  cnptoprest  14475  txcnp  14507  txcnmpt  14509  metcnp3  14747